Analytical and Numerical Experimentation with a Low-Order Population-Energy Model

NOT FOR QUOTATION WITHOUT PERMISSION OF THE AUTHOR

ANALYTICAL AND NUMERICAL

EXPERIMENTATION WITH A LOW-ORDER POPULATION-ENERGY MODEL

A. A. Harms July 1981 WP- 81-95

Working Papers are interim reports on work of the International Institute for Applied Systems Analysis and have received only limited review. Views or opinions expressed herein do not necessarily repre- sent those of the Institute or of its National Member Organizations.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS A-2361 Laxenburg, Austria

ABSTRACT

An experimental analysis is undertaken to formulate and examine a tractable national population-energy model. On the basis of plausibility arguments a mathematical representation is established that serves to describe population and per- capita energy consumption in a coupled dynamic relationship.

The conceptual basis of the model is tested numerically with the help of historical data on distinct national population- energy patterns. The structural form of the model and its limitations are further examined and extensions are suggested.

Analytical and .Numerical Experimentation With A Low-Order Population-Energy Model

1. INTRODUCTION.

The formulation of analytical models in IIASA's Energy Systems Program attempting to characterize patterns of the population and energy relationship has been the objective of several studies (Hafele 1976; Hafele and Burk 1976; Griimm and Schrattenholzer 1976; Breitenecker and Griimm 1981). These in- vestigations try to both simulate and represent specific

relationships involving national parameters--such as popula- tion, energy consumption, labor input, gross domestic product, etc.

The present investigation endeavors to continue that line of analysis but sets out from an entirely different starting point. By way of analytical manipulation based on a well-

accepted methodology, it develops a phase-space representation that formally corresponds to previous IIASA studies but also incorporates empirical data.

In the following, we first proceed to formulate a plausi- bility representation for the dynamics of the population-

energy interaction and then employ empirical data to deter- mine specific numerical-graphical representations. Finally, we suggest some extensions to this model and comment on some details of the implicit structural characteristics of the model.

2. ANALYTICAL-PLAUSIBILITY FORMULATION.

We consider a nation--or any other suitable region--that is characterized by a time-dependent population P(t) and a

time-dependent per-capita energy consumption q(t)

.

^We

accept the premise that these two functions are coupled in a manner which is case-specific and may vary with time. The form and strength of the population-energy coupling is taken to be determined by the specific mathematical form and the magnitude

(as well as sign) of appropriate coefficients.

In order to establish the mathematical model of interest here, we use the following plausibility argument applicable to a finite, though arbitrary, time interval At

.

^{We hypo-}

thesize that the fractional change in the per-capita energy consumption over time interval At possesses two basically different components: one that is essentially constant and another that is linearly dependent on the population. That is, the proportionality relationship using superposition is written as

* a [ k + P ( ~ ) I over ~t

,

q (t)

with k being a constant.

Equation (1) can b= better understood by way of Fig. 1 which illustrates two distinct linear variations of Aq/q with P

over the time interval At

.

The line with a positive slope may, for example, be associated with the determination and

resourcefulness of a growing population to provide an increasing per-capita energy accessibility; the negative slope, however, implies a diminishing per-capita energy consumption with an increasing population suggesting, for example, that the total energy resource is essentially constant and must be distributed among more people. Differences between the trends are under- _, stood to stem from dissimilarities in the technological develop- ment and resource situation of the countries considered.

By translating this qualitative argument into mathematical terms, the proportionality relationship of Equ.(l) can be

I I

I Expansive

I

[<

E z i b i l i t y

I I

\

^Energy

Resource Depletion

POPULATION, P(t)

Figure 1: Schematic illustration showing hypothesized relationship of a fractional per-capita energy change as a function of population during time interval At. A conceivable explanation for the trend is indicated.

e x p l i c i t l y w r i t t e n a s a f i n i t e d i f f e r e n c e e q u a t i o n

where p1 and p 2 a r e s p e c i f i c c o n s t a n t s . T a k i n g t h e l i m i t

A t + O and a l l o w i n g t t o b e a r b i t r a r y one o b t a i n s t h e d i f f e r - e n t i a l e q u a t i o n

While t h i s e q u a t i o n h a s a u s e f u l form, i t r e p r e s e n t s o n l y o n e d i f f e r e n t i a l r e l a t i o n o f two f u n c t i o n s q ( t ) and P ( t )

.

^{W e} t h e r e f o r e p r o c e e d t o e s t a b l i s h a c o m p a r a b l e p l a u s i - b i l i t y argument f o r P ( t ) a s a f u n c t i o n o f q ( t )

.

C o n s i d e r i n g t h e f r a c t i o n a l change i n p o p u l a t i o n w e p o s t u - l a t e t h a t i t i s a l s o r e l a t e d t o t h e p e r - c a p i t a e n e r g y consump- t i o n i n a l i n e a r manner:

a AP

P ( t ) [ k + q ( t ) l o v e r t . ( 5 )

The p o s l t i v e s l o p e i n F i g . 2 s u g g e s t s t h a t , f o r example, i n - c r e a s i n g e n e r g y consumption l e a d s t o improved h e a l t h c a r e , which r e d u c e s i n f a n t m o r t a l i t y and e x t e n d s l i f e e x p e c t a n c y . On t h e o t h e r hand, t h e f a l l i n g s l o p e may imply a d e l i b e r a t e b i r t h r a t e r e d u c t i o n i n o r d e r t o i n c r e a s e t h e s t a n d a r d o f l i v i n g by i n c r e a s e d e n e r g y consumption. W e p r o c e e d t o w r i t e t h e r e f o r e

I

:

I

f

Expanded public services associated with increasing energy consumption and

t

I population growth

Decreasing birth rate associated with trends towards energy- intensive

standards of living

PER-CAPITA ENERGY, q(t1

Figure 2: Graphical depiction of hypothesized relationship between a fractional population change as a

function of per-capita energy consumption during time interval At. A plausible explanation for such linear trends is indicated.

from which, by the analogous argument used to obtain Equ.(4), we obtain

Equations (4) and (7) define the mathematical model that describes the energy-population pattern. They possess a

symmetry which becomes more obvious if they are written in a general form for arbitrary functions x (t) and y (t) :

and

with no apparent restriction on the sign or magnitude of coefficients alt a2t bl and b2

.

3. COMMENT ON THE DYNAMICAL EQUATIONS.

Equations (4) and (7) represent a system of coupled

dynamical equations whose form are known to appear in various contexts (Kemeny and Snell 1962; Haken 1977). Of interest to us here are certain aspects associated with the coefficients alf a2t lJ1 and p2

.

In view of the formulation of the problem, there is no physical reason to expect any restrictions on the signs of the coefficients; both populations and per-capita energy consump- tion can increase or decrease with time resulting therefore in positive and negative signs as appropriate.

If Equs. (4) and (7) constitute an acceptable representa- tion of "reality" then the coefficients are, in a complex way, functions of the various pertinent independent para- meters such as birth rates, death rates, gross national

product, standard of living, cost of capital, type of economy,

and many others. This would then suggest that the coefficients

0 2 t Pl and p2 might best be studied by fitting Equs.(4)

and (7) to specific regional data. Further, since these determining factors change with time, these coefficients are likely to vary with time,

The space and time dependency of coefficients ult 0 2 t

P1 and P2 suggests that careful attention be paid to the applicability of the equations. The space dependence is easily accommodated by calculating these coefficients

for each region of interest. The time dependence imposes the restriction that, once the coefficients are known, the dynamical equations are useful as predictive tools only if the coefficients do not vary with time; otherwise, only a descriptive property can be associated with the mathematical characterization.

The above points suggest a potential domain of application of the dynamical equations, Equs

.

(4) and (7)

,

under conditions that the coefficients can be accurately calculated; it refers to the identification of trajectory tendencies for the national population-energy patterns and indicates the intrinsic (geo- metric) structure characterizing the pattern of population and energy.

The establishment of phase-space representations, as suggested in.previous IIASA studies quoted above, is parti- cularly relevant in this regard,

4. PHASE-SPACE CHARACTERIZATION,

To obtain the phase-plane representation and the associa- ted trajectory, we eliminate time from Equs. (4) and (7) by division

and identify the pertinent equilibrium point on the P-q phase plane by the coordinates

The solution of Equ.(9) can be deduced and, for the case that the signs of al, 02, p 1 and

u2 are arbitrary, is given by the transcendental equation

Here C is an integration (i.e., contour) constant and S(z) = sign ( 2 ) x 1(=+1)

.

The resultant phase-space surfaces, or phase-portraits, can be conveniently characterized by the sign of the ratios of the coefficients:

(

^{< 0 , elliptic}

u

¹

'

1 i =

1

^> 0 , hyperbolic

Figure 3 provides one illustration, a hyperbolic phase- space portrait in this case, for which the equilibrium points exist in the positive P (t) and positive q(t) quadrant.

Note the characteristic trajectories in the four domains and the Pattern of the separatrix. Other substantially different P-q portraits are mathematically feasible as we will show.

5. NUMERICAL EVALUATION.

To assess the adequacy of the formulation for our pur- poses here and to determine whether or not some useful para- meterization might result, we undertook to use historical data for the determination of the coefficients as temporal

Equilibrium point

POPULATION, P

Figure 3: Example o f trajectories and domains o f the P-q phase portrait; other portraits are possible.

averages for countries which seemed to represent sufficiently diverse population-energy characteristics: Canada, Czecho- slovakia, India, Nigeria, Sweden, USA, and USSR. The

population data, P(t)

,

and the per-capita energy consumption data in units of tons-equivalent-of-coal, q(t)

,

^{for the}

period 1957-66 (World Energy Supplies 1976) were selected;

since this period seemed generally free from large-scale perturbation, the calculation of such temporal-average co- efficients al, a2, p1 and

3

appeared appropriate.

However, the apparent noise in the q(t) data compli- cated the determination of the coefficients al, a2, p1 and lJ2 as well as identification of the phase-space portrait.

To eliminate this noise effect we smoothed the raw historic P(t) and q(t) data by fitting least squares polynomials of the form

and used the resultant values to obtain least-squares fits for alt a2t lJ1 and to Equs. (4) and (7)

,

which were written in the linear form

dP(t) = a

+

^{a2q (t)}

P(t) dt 1 and

Figure 4 displays a typical result showing both the raw data and the smoothed data.

This procedure yielded excellent fits with the

correlation coefficients well in excess of 0.8 with typical values of 2.99; Table 1 lists coefficients, correlations, equilibria points, and the geometric phase-space feature.

I I I I I ^{1 I I I} I

1957 1966

TIME, (year)

Figure 4: Population and energy data for Canada (1957-1966).

The solid line refers to raw data (-) and the dashed line identifies the smoothed results ( - - - ) ;

the population raw and smoothed data are essentially identical.

. . .

d d p o o o o I

O O \ b N o \ U b m m m Q ) m m

.

m

.

d d d d d o o

I I I 1

6. ILLUSTRATION OF RESULTS.

The phase-space portraits for the seven countries and for the 10-year period 1957-66 are shown in Figs. 5 to 11. Note again that if the parameters ol, 02, p1 and p2 do not change with time the specific phase-portrait is stationary and the evolution of the population and per-capita energy pattern is thus fully prescribed. These patterns are charac- terized by elliptic and hyperbolic geometrics about the

equilibrium point suggesting either cyclic or asymptotic

trajectory tendencies. Note also the considerable distortion of these two classes of surfaces, especially in the two

cases--India and Nigeria--where in reality equilibria points would be unattainable.

Interestingly, as displayed in Table 2, the cype of trajec- tory and location of the equilibrium point can be associated with a particular type of resource economy. While the results

exhibit interesting and distinct features of the different national energy-population patterns, they highlight in parti- cular the temporal average state of the 10-year period 1957-66.

An assessment of how the P-q surfaces vary with time is clearly of interest and is examined below.

7. TEMPORAL VARIATIONS.

The temporal variations of the population-energy surface have been assessed for .the example of Canada. The relevant population-energy data (United Nations 1976 and 1979) for the 29-year period 1950-1978, are displayed in Fig. 12. As indi- cators of the temporal variation of the P-q surface we have chosen are the history of the equilibrium point (PE, qE) and the characteristic geometric shape--either hyperbolic or elliptic--for successive 10-year sliding intervals. That is, for each of the time-intervals 1950-59, 1951-60, 1952-61, 1953-62,

...,

1969-78, we calculated ol, 02,

ul

and p2;

from this we obtained the equilibrium point, Equ.(lO), and identified the geometric shape of the corresponding (P-q) surface, according to Equ. (12)

.

The polynomial data smoothing procedure used previously was also applied to this part of the analysis.

I CANADA

14 16 18 20 22

POPULATION, P (10')

Figure 5: Population-energy phase-portrait and trajectory, Canada, 1957-1966.

CZECHOSLOVAKIA

12 13 14 15

POPULATION, P (lo6)

Figure 6: Population-energy phase-portrait and trajectory, Czechoslovakia, 1 9 5 7 - 1 9 6 6 .

300 400 500 600

POPULATION, P

(lo6)

Figure 7: Population-energy phase-portrait and trajectory, India, 1957-1966.

I

^NIGERIA

40 50

POPULATION,

P (1O81

Figure 8: Population-energy phase-portrait and trajectory, Nigeria, 1957-1966.

6

^p

¹⁵⁰

CC

CT

>=

C3 a

W

5 loo a k

2

0

a

UJ

50 e

SWEDEN

-

5.0 6.0 7.0 8.0

POPULATION, P (lo6)

Figure 9: Population-energy phase-portrait and trajectory, Sweden, 1957-1966.

I USA

140

160 180 200 220

POPULATION, P (10')

Figure 10: Population-energy phase-portrait and trajectory, U.S.A., 1957-1966.

CANADA

14 16 18 20 22 24

POPULATION, P (lo6)

Figure 12: Historical population and per capita energy consumption, Canada, 1950-1978.

The resultant trajectory of the equilibrium point

(P,I qEIi for the i = 1,2,

...,

20 sliding overlapping time intervals is displayed in Figure 13.

The-equilibrium trajectory in the Figure clearly dis- plays some dynamic transformations and strong resonance

features. The geometry starts out as a hyperbola, (1950-59), then transforms itself into an ellipse (1951-60), and returns to a hyperbola for the next two sliding time periods. From 1954-63 to 1960-69 it remains an ellipse, transforms to a hyperbola in 1961-70, and remains in that shape for the

remainder of 'the study period. The other interesting feature is the apparent tendency to seek what may be termed resonances of its equilibrium coordinates PE or for single isolated intervals. Some simplification of the complexity of Fig. 13 is possible by eliminating the "exceptional" years 1951, 1956, and 1975; the clear separation of the elliptic and hyperbolic domains (Fig. 14) is the interesting feature.

8. POTENTIAL EXTENSIONS.

The preceding suggests that the low-order population-

energy model formulated and tested here can be used to establish

"temporal-average" phase-space portraits for a specified historical period. As is shown in the analysis leading to Figures 13 and 14, these "average" portraits can undergo

dramatic transformations over time. The appearance of physically unattainable equilibria points is a particularly interesting

aspect for it may well tell us something about the model as well as about certain features of the national economies considered.

One possible and pertinent use of this methodology could be a comparative analysis of national population-energy

patterns for specified time intervals using the temporal average portr~its as indicators. The distinction between

"prediction" and "descriptions" seems most pertinent here.

$

^Extremum

(qE =-4 x 10')

-40) ^{I I} I I I

10 15 20

25 30 35

POPULATION EQUILIBRIUM, PE (10'1

Figure 13: Illustration showing movement of equilibrium point (PE,qE) based on 10-year sliding period.

The open circles characterize elliptic P-q phase-portraits while the solid circles characterize hyperbolic P-q phase-portraits.

Data is for Canada, 1950-1978.

Ellipse Domain Hyperbola Domain

I

POPULATION EQUILIBRIUM, PE(106)

Figure 14: Movement o f equilibrium point ( P E t qE) with the "exceptional" years 1951, 1956, and 1975 removed.

More substantially, the phase-portraits might be used to identify trajectories of future national trends and tendencies.

Two ways seem possible. One particular approach would be to obtain a coefficient-mapping for different societal domains by way of characterizing the ccefficients--which are evidently slow functions of time--in terms of other determining factors.

The other approach would involve weighting of the historical data with more weight on more recent data and less weight on more distant historical information.

Finally, one may wish to remove the linearity imposition of Equ. (1) and Equ. (5) and add higher order terms recognizing, however, that the coefficients added may introduce a further degree of complexity.

9. CONCLUDING COMMENT.

It appears that the dynamics of national population-energy characteristics can be investigated by using methodologies

that are based on the formulation of coupled low-order equa- tions. The model developed and the results established here suggest some new directions for further research.

ACKNOWLEDGEMENT

This report was written during the authors affiliation as Associated Research Scholar with the International Intitute for Applied Systems Analysis, Laxenburg, Austria. The author

gratefully acknowledges the hospitality'he received during his stay at IIASA as well as the discussions he had on the subject of this paper with Dr. John Casti, Dr. Manfred Breitenecker and Dr. Zenon Fortuna. The calculatfonal work displayed were undertaken by Dr. W. M. Krenciglowa.

REFERENCES

Hafele, W. (1976) "Resilience of Energy Systems" in Second Status Report of the IIASA Project on Energy Systems by W. Hafele etal, RR-76-1. Laxenburg, Austria:

International Institute for Applied Systems Analysis, Hafele, W, and R. Burk (1976) An Attempt of Long-Range Macro-

economic Modelling in View of Structural and Technologi- cal Change, RM-76-32. Laxenburg, Austria: International Institute for Applied Systems-Analysis.

Grdmm, H. R. and L. Schrattenholzer (1976) Economy Phase

Portraits, RM-76-61. Laxenburg, Austria: International Institute for Applied Systems Analysis.

Breitenecker, M. and H. R. Grfimrn (1981) Economic Evolutions and Their Resilience: A Model, RR-81-5. Laxenburg, Austria:

International Institute for Applied Systems Analysis.

Kemeny., J. G. and J. L. Snell (1962) Mathematical Models in the Social Science. Waltham, Massachusetts: Blaisdell Pub- lishing Co.

Haken, H. (1977) Synerqetics. Berlin: Springer-Verlag.

United Nations (1976) World Energy Supplies 1950-1979, Statis- tical Paper Series, Series J, No. 19. New York: United Nations.

United Nations (1979) World Energy Supplies 1973-1978,

Statistical Papers, Series J , No. 22. New York: United Nations.